In mathematics, a prime number (or prime) is a natural number that has exactly two distinct positive divisors, 1 and itself. A composite number is a natural number that has more than two positive divisors.

The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…

The first few composite numbers are 4, 6, 8, 9, 10…

Prime numbers have fascinated mathematicians since ancient times. The Greek mathematician Euclid proved that there were infinitely many prime numbers. It is not known if there are any odd perfect numbers (a number that is the product of its own positive divisors, excluding the number itself), but it is known that there are infinitely many perfect numbers and infinitely many odd composite numbers. An infinite number of primes can be proved using the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of prime numbers.

**How many factors does 13 have?**

13 has two factors, 1 and 13.

**How to find if a number is prime or composite?**

To test if a number is prime, you can use either of these methods.

METHOD 1: Make a list of the factors of the number n. If any one of them divides into n evenly then it is not prime. That means that there are numbers smaller than itself which divides into it without leaving a remainder. For example let us take the number n= 8. The list of all factors of 8 are 1, 2, 4 and 8. So we can see that 4 divides into 8 without leaving a remainder and so 8 is not prime.

METHOD 2: Use the Sieve of Eratosthenes to find all the multiples of a number smaller than n. Cross out all the multiples of that number on the list and then test to see if any remain. For example, let us take the number n= 15. The multiples of 3 are 3, 6, 9, 12, 15. So we cross out all these numbers on the list of factors and then test to see if any remain. The factors of 15 are 1, 3, 5, 15. So we see that none of the remaining numbers divides into 15 without leaving a remainder. Thus 15 is prime.

The sieve of Eratosthenes: Start at 2 and cross out every second number as follows: 4, 6, 8…18, 20.

Then starting at 20 and continuing upwards, cross out every third number as follows: 30, 40…90.

At 60 we have the first prime and in fact all others up to 100 will be crossed out because they are divisible by 2 or 3.

Thus the primes from 2 to 100 are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37…

**How to determine a prime number?**

To determine a prime number we can use the Sieve of Eratosthenes to list out all the factors of a given number. If any factor divides into that number without leaving a remainder then it is not prime.

The Fundamental Theorem of Arithmetic: Every positive integer greater than 1 can be written as a product of one or more prime numbers. This theorem is the basis for determining if a number is prime or composite.

For example, let us take the number 567. We can see that it is not prime because it can be written as a product of two prime numbers, 3 and 191. 567= 3×191.

**What are some applications of prime numbers?**

Prime numbers have a number of applications in mathematics and beyond. For example, they can be used to create secure codes and to encrypt data. Prime numbers are also used in mathematical problems and puzzles, such as the Sieve of Eratosthenes. Finally, there is an ongoing search for new prime numbers that has led to the discovery of some very large prime numbers.

In cryptography, prime numbers are used in two different ways. The first way is to create a public key and private key. The public key, made up of the product of two prime numbers, is known by everyone. However, only the person who has the private key can read messages encrypted with it.

**Conclusion**

Prime numbers are very important in mathematics and have a number of interesting applications. They can be used to create codes and solve mathematical problems. The search for new primes has led to the discovery of some very large prime numbers.